Quantum spin liquid and electric quadrupolar states of single crystal Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$

The ground states of the frustrated pyrochlore oxide Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$, sensitively depending on the small off-stoichiometry parameter $x$, have been studied by specific heat measurements using well characterized single crystalline samples. Single crystal Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ boules grown by the standard floating zone technique exhibit concentration ($x$) gradient. Values of $x$ of small crystal samples cut from a boule are estimated by precisely measuring the lattice constant. The ordered phase of electric quadrupole moments shows a dome structure in the $x-T$ phase diagram with the highest $T_{\text{c}} \simeq 0.5$ K at $x \simeq 0.01$. This phase diagram suggests that the putative U(1) quantum spin liquid state of Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ exists in the range $x